Mr. Trump acknowledged his loss in a video Thursday night while saying he was outraged by the “lawlessness and mayhem” at the Capitol. “Tempers must be cooled,” Mr. Trump said, a day after he had encouraged his supporters to march on Congress because of his false allegations that fraud had denied him re-election.

On Thursday, Transportation Secretary Elaine Chao and Education Secretary Betsy DeVos said they were resigning their posts with just under two weeks left in the administration, with Ms. DeVos saying in a letter, “There is no mistaking the impact your rhetoric had on the situation.” Mick Mulvaney, a former White House chief of staff who is now U.S. special envoy for Northern Ireland, quit as well.

The developments highlighted the alarm Wednesday’s riot set off and the belief among many lawmakers from both parties that Mr. Trump, a Republican, provoked it and didn’t condemn the actions strongly enough once the protests turned violent. On Wednesday night, with Congress reconvened after the riot, Senate Majority Leader Mitch McConnell (R., Ky.) called the attack a “failed insurrection.”

Vice President Mike Pence ultimately confirmed Mr. Biden, a Democrat, as the election winner early Thursday morning.

The head of the Capitol Police and the sergeants at arms for the House and Senate resigned after lawmakers demanded accountability from officials responsible for the security of the complex. Congressional leaders from both parties and both chambers said they had started an investigation into why police were quickly overwhelmed, letting rioters roam through the building and forcing Congress to delay its count of states’ results.

The Capitol Police said some officers were hospitalized as a result of the melee, and one officer, Brian D. Sicknick, died Thursday night from injuries suffered “while physically engaging with protesters.” Four Trump supporters also died during the riots, including a woman who was shot in the Capitol building.

Hundreds of rioters, pushing past security barriers and Capitol Police officers, breached the Capitol on Wednesday afternoon as both the House and Senate were meeting inside, forcing the evacuation of some lawmakers, while others barricaded themselves against the crowds.

The Justice Department said 15 criminal complaints would be filed Thursday in connection with the riot, and the Federal Bureau of Investigation was seeking the public’s help in identifying more suspects.

Former Attorney General William Barr, who resigned just before Christmas, said that Mr. Trump’s conduct on Wednesday was a “betrayal of his office and supporters.”

A number of White House officials had resigned Wednesday. Among them were Matthew Pottinger, the deputy national security adviser, and Stephanie Grisham, chief of staff and press secretary to first lady Melania Trump.

After Mr. Pence, presiding over a joint session of Congress that was interrupted for hours due to the rioters, confirmed Mr. Biden as the winner early Thursday morning, Mr. Trump released a statement at 3:49 a.m. saying he “totally” disagreed with the outcome but “nevertheless there will be an orderly transition on January 20th.”

“While this represents the end of the greatest first term in presidential history, it’s only the beginning of our fight to Make America Great Again!” he said in the statement, which was released through an aide as the president’s Twitter account was locked at the time due to postings Wednesday the social media network said could spread violence.

Twitter said the president would be allowed to resume tweeting on Thursday, but Facebook and Instagram said Mr. Trump would be blocked indefinitely, saying the risks are “simply too great.”

Hundreds of rioters, pushing past security barriers and Capitol Police officers, breached the Capitol on Wednesday afternoon as both the House and Senate were meeting inside, forcing the evacuation of some lawmakers, while others barricaded themselves against the crowds.

Mr. Biden characterized the mob that stormed the U.S. Capitol on Wednesday as domestic terrorists. “Don’t dare call them protesters,” Mr. Biden said in remarks from Wilmington, Del. “They were a riotous mob. Insurrectionists. Domestic terrorists. It’s that basic. It’s that simple.”

Mr. Schumer, poised to become the Senate majority leader when Democrats take control of the chamber from Republicans later this month, said that Mr. Pence and the cabinet should work to remove Mr. Trump by invoking the 25th amendment.

Some administration officials have had preliminary discussions about the 25th amendment, according to a senior administration official and others familiar with the matter. However, any move to remove Mr. Trump is unlikely to happen so close to the inauguration, and neither Mr. Pence nor any cabinet members have expressed support for such a move. Impeachment of Mr. Trump, for what would be the second time, is also unlikely as Congress is out of session.

“I think we have to hold our breath,” until Mr. Biden’s inauguration on Jan. 20, said Sen. Mitt Romney (R., Utah), saying it was too late to begin proceedings to remove the president.

A number of other Democrats also called for the move, and one Republican, Rep. Adam Kinzinger of Illinois, joined them on Thursday.

“Here’s the truth: the president caused this. The president is unfit and the president is unwell. And the president must relinquish control of the executive branch voluntarily or involuntarily.”

Under Siege

An ordinarily uneventful Congressional proceeding—the recording of electoral-college votes—descended into violent tumult Wednesday.

The 25th amendment says that if the vice president and a majority of the officers of the sitting cabinet secretaries decide that the president is unable to do the duties of his office, the vice president can immediately take on the duties of the acting president. To officially remove the president, Congress would then vote, and two-thirds of each chamber would be required to officially remove the president. If that threshold isn’t reached, the president resumes office.

Congressional leaders expressed alarm at how quickly the Capitol was infiltrated. By the end of Thursday, the three top security officials in the Capitol had resigned. Mr. McConnell called the breach a “massive failure” for the Capitol’s security and accepted the resignation of Senate Sergeant at Arms Michael Stenger. Mrs. Pelosi said that House Sergeant at Arms Paul Irving would also be resigning after Wednesday’s events.

Mrs. Pelosi also called for the resignation of Capitol Police Chief Steven Sund. He submitted his resignation late Thursday.

The union representing Capitol Police officers had also called for a leadership change, criticizing the lack of preparation.

“Our officers are experienced and they are dedicated, but they lacked the immediate backup and equipment needed to control the surging crowd,” said union leader Gus Papathanasiou.

A Capitol Police officer shot the woman who later died of her injuries.

Other cabinet officials expressed concerns about the violence. Acting Homeland Security Secretary Chad Wolf called the actions of the rioters “unacceptable” and urged Mr. Trump to condemn the violence. He said he would remain in his position until the end of the administration.

National-security adviser Robert O’Brien was considering resigning but hadn’t made up his mind to do so, according to people familiar with the situation. Republican allies in Congress also criticized the president’s actions and his rhetoric claiming a rigged election.

“Millions of Americans were misled on this vote,” said South Carolina Rep. Nancy Mace, a freshman Republican who campaigned with the president. Mr. Trump, she said, “went too far” on his rhetoric and it led to violence.

“The president needs to understand that his actions were the problem not the solution, that the rally yesterday was unseemly, it got out of hand,” said Sen. Lindsey Graham (R., S.C.).